NPER

NPER, or number of periods, is a financial concept used to determine how many payment periods are required for an investment or loan to reach a specified financial outcome. It is a standard input or result in financial calculators and spreadsheet functions dealing with loans, mortgages, and savings plans.

NPER is found by solving the cash flow equation that relates present value, future value, periodic payment, and the per-period interest rate. The general equation is FV = PV*(1+i)^n + PMT*[(1+i)^n - 1]/i, where i is the interest rate per period, n is the number of periods, PV is the present value, PMT is the payment per period, and FV is the desired future value. In simple cases with FV = 0 and payments at the end of each period, there is a closed-form solution: n = -ln(1 - i*PV/PMT) / ln(1+i). When FV is nonzero or payments occur at the beginning of periods, solving for n typically requires numerical methods.

Inputs and sign conventions: i is the rate per period; PV and PMT are entered with consistent signs (for example, PV positive when receiving money and PMT negative when paying). The type parameter indicates whether payments are due at the end of a period (0) or at the beginning (1).

Applications and limitations: NPER is used to estimate the duration of loans, mortgages, and savings plans under fixed payments and rates. It assumes a constant interest rate and a fixed payment schedule; results can be non-integer and depend on sign conventions. Real-world loans with fees or variable rates may require adjustments.

See also: RATE, PMT, PV, FV, financial calculator, Excel's NPER function.